Help with Physics Problem: Finding Distance d for Ball to Strike Peg

[chuxingli]

the question:

You have a supporting point and a ball connected by a
string of length 100cm. There is a peg at a distance d
below the supporting point. You release the ball from
rest at an instance when the ball is totally
horizontal to the point, it swings along an arc. When
the string catches on the peg, the ball swings upward
but does not complete a circular path around the peg.
Instead, the abll leaves its circular path, and
strikes the peg.

1) Find the distance d below the point of support at
which the peg should be placed for this to happen.

2) You have to find a function d(theta) at which the
peg needs to be placed for this to happen.

 

[Integral]

Perhaps the reason this question has not been replied to is that the final part is not clear. How does the ball strike the peg? Could you post a picture showing the situation?

 

[lightgrav]

This is a tough problem.

Finding the speed of the ball at the bottom is easy (use E conservation).

after string hits peg, the ball motion is circular up past horizontal,
but you don't know the effective length of the string L = (1[meter] - d).

Call theta the angle above the horizontal at which ball leaves the circle.
ball is a 2-d projectile after that, from (x = L cos theta , y = L sin theta)
with velocity components (- v sin theta , v cos theta) , to peg at (0,0),

This is much easier problem in the forward direction (given d, where does ball cross x=0 line) . maybe you program it into a calculator or spreadsheet, then change d until it hits peg ...?

 

[plusaf]

i agree with Integral...

the question:
You have a supporting point and a ball connected by a
string of length 100cm. There is a peg at a distance d
below the supporting point. You release the ball from
rest at an instance when the ball is totally
horizontal to the point, it swings along an arc. When
the string catches on the peg, the ball swings upward
but does not complete a circular path around the peg.
Instead, the ball leaves its circular path, and
strikes the peg.
1) Find the distance d below the point of support at
which the peg should be placed for this to happen.
2) You have to find a function d(theta) at which the
peg needs to be placed for this to happen.

wouldn't the diameter of the ball and the peg matter?
i can picture a solution with peg of 1 cm diameter and ball of 2 cm diameter, and the peg 96 cm vertically below the "supporting point"...

 

[chuxingli]

plusaf, the peg and ball are considered points

 

[chuxingli]

there are 3 concepts involved:

1. conservation of energy at the point when the ball leaves the smaller circle

2. speed at the point when the ball leaves the smaller circle.

3. freefall.

can someone who knows physics work it out for me. the concepts are quite simple. But the algebra i did was extremely messy.

 

[chuxingli]

Integral, obviously, there is no outside fore so there is only one way the ball could hit the peg...

 

[lightgrav]

Did you define the angle above the horizontal that the circle becomes parabola?

you should know the speed there will depend on L and d.
I think that 2/3 d = L sin theta , and v^2 (at transition) = L g sin theta .
I had to solve a quadratic in sin^2 theta , to hit the peg.
So there are two angles, for a 1-meter string, if I've not made a sign mistake.
(yeah, the algebra stinks. Some problems are best done on computer ...
say, do you have access to Maple ...?)

 

[beluluk]

i'm on it

after string hits peg, the ball motion is circular up past horizontal,
but you don't know the effective length of the string L = (1[meter] - d).
Call theta the angle above the horizontal at which ball leaves the circle.
ball is a 2-d projectile after that, from (x = L cos theta , y = L sin theta)

agree

with velocity components (- v sin theta , v cos theta) , to peg at (0,0),

somehow i couldn't agree. i imagined the ball stopped vertically at one point above the peg. and that makes the velocity components (vy=0, vx= something). and from that, i worked out that
d=100 - {(2*vx^2*sin(theta))/([cos(theta)]^2*g)}

i'm working the vx. I'm trying to understand how the energy conservation work. when the vertical component of velocity is zero, the horizontal is not.

 

[beluluk]

ehm... when i say theta, i mean
"the angle between the horizontal line along the peg and the point where the ball starting to leave the circular path.
and also, i was assuming that the peg was "straight" below the supporting point

 

[bjd40@hotmail.com]

the peg and the ball problem

First let us imagine the string does not break(it is not mentioned in the problem, but if u imagine this it is relatively simpler). now at certain angle above the horizontal this happens. the angle can be foundby equating the gravity force (downward) and the upward component of mv^2/r at that point. also v can be calculated from energy considerartion. Now treating it as a projectile problem the hitting of the peg can be understood. you can try this way. I have not done thew mathematics myself. So, i don't know how rigorous it is.